A102002 Weighted tribonacci (1,2,4), companion to A102001.

1, 7, 13, 31, 85, 199, 493, 1231, 3013, 7447, 18397, 45343, 111925, 276199, 681421, 1681519, 4149157, 10237879, 25262269, 62334655, 153810709, 379529095, 936489133, 2310790159, 5701884805, 14069421655, 34716351901, 85662734431, 211373124853, 521564001319

Offset

1,2

Comments

a(n)/a(n-1) tends to 2.46750385...an eigenvalue of M and a root of the characteristic polynomial x^3 - x^2 - 2x - 4. A102001 is generated from [1 1 1 / 2 0 0 / 0 2 0] but has the same characteristic polynomial and recursive multipliers (1,2,4). A101000 uses the recursive multipliers (1,2,4,8).

Links

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (1,2,4).

Formula

a(n) = a(n-1) + 2*a(n-2) + 4*a(n-3), a>3. a(n) = center term in M^n * [1 1 1], where M = the 3X3 matrix [0 1 0 / 0 0 1 / 4 2 1]; M^n * [1 1 1] = [a(n-1) a(n) a(n+1)].

G.f.: -x*(4*x^2+6*x+1)/(4*x^3+2*x^2+x-1). [Harvey P. Dale, Apr 28 2012]

Example

a(6) = 199 = 85 + 2*31 + 4*13 = a(5) + 2*a(4) + 4*a(3).
a(6) = 199 since M^6 * [1 1 1] = [85 199 493] = [a(5) a(6) a(7)].

Mathematica

LinearRecurrence[{1, 2, 4}, {1, 7, 13}, 50] (* Harvey P. Dale, Apr 28 2012 *)

Prog

(Sage)
from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(1, 1, 1, 1, 2, 4)
[next(it) for i in range(32)]
# Zerinvary Lajos, Jun 25 2008

Crossrefs

Cf. A102000, A102001.

Sequence in context: A342691 A215932 A091432 * A239679 A195353 A342184

Adjacent sequences: A101999 A102000 A102001 * A102003 A102004 A102005

Keyword

nonn,easy

Author

Gary W. Adamson, Dec 23 2004

Extensions

More terms from Harvey P. Dale, Apr 28 2012

Status

approved